Why are all these different colored lines showing up in Edit Mode?
The white lines are most likely edges, that have been "marked as sharp". 

This happens quite often, when importing objects from other file formats.

Select your mesh, go into Edit Mode, hit A to select everything, then hit Ctrl+E and select "Clear Sharp" in the popup menu. 



There also might be double vertices in your mesh (hard to say from the image alone). 

To remove double vertices, select your mesh, go into edit mode, hit A to select everything, then hit W and select "Remove Double Vertices" in the popup menu.